Pause in relationships and constraints
It doesn't collect all the transforms, it just uses the relative offset, that the target had have at the moment you created the constraint. For the Child of there are two buttons 'Set inverse', and 'Clear inverse'. They change that stored offset. So to get what you ask, you need to change the offset. I have no idea, where and how it is stored in the constraint data, but I'm sure it's accessible from the python API.